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Higher order numerical schemes for SPDEs with additive Noise

Abhishek Chaudhary, Andreas Prohl

TL;DR

The paper develops high-order numerical schemes for linear SPDEs with additive noise by transforming the SPDEs into random PDEs and applying a modified Crank–Nicolson approach. For the stochastic heat equation, the scheme achieves a strong convergence rate of $\mathcal{O}(\tau^{3/2})$ under sufficiently smooth data, while for the stochastic wave equation the corresponding scheme attains a rate of $\mathcal{O}(\tau^{2})$. These results rely on enhanced temporal regularity of the transformed solutions and a hybrid quadrature strategy that treats deterministic and stochastic integrals on appropriately refined meshes. Numerical experiments on $D=(0,1)$ corroborate the theoretical rates, demonstrating substantial improvements over standard Euler-type schemes and validating the practicality of the proposed methods for general elliptic operators $A$.

Abstract

We present high-order numerical schemes for linear stochastic heat and wave equations with Dirichlet boundary conditions, driven by additive noise. Standard Euler schemes for SPDEs are limited to an order convergence between 1/2 and 1 due to the low temporal regularity of noise. For the stochastic heat equation, a modified Crank-Nicolson scheme with proper numerical quadrature rule for the noise term in its reformulation as random PDE achieves a strong convergence rate of 3/2. For the stochastic wave equation with additive noise a corresponding approach leads to a scheme which is of order 2.

Higher order numerical schemes for SPDEs with additive Noise

TL;DR

The paper develops high-order numerical schemes for linear SPDEs with additive noise by transforming the SPDEs into random PDEs and applying a modified Crank–Nicolson approach. For the stochastic heat equation, the scheme achieves a strong convergence rate of under sufficiently smooth data, while for the stochastic wave equation the corresponding scheme attains a rate of . These results rely on enhanced temporal regularity of the transformed solutions and a hybrid quadrature strategy that treats deterministic and stochastic integrals on appropriately refined meshes. Numerical experiments on corroborate the theoretical rates, demonstrating substantial improvements over standard Euler-type schemes and validating the practicality of the proposed methods for general elliptic operators .

Abstract

We present high-order numerical schemes for linear stochastic heat and wave equations with Dirichlet boundary conditions, driven by additive noise. Standard Euler schemes for SPDEs are limited to an order convergence between 1/2 and 1 due to the low temporal regularity of noise. For the stochastic heat equation, a modified Crank-Nicolson scheme with proper numerical quadrature rule for the noise term in its reformulation as random PDE achieves a strong convergence rate of 3/2. For the stochastic wave equation with additive noise a corresponding approach leads to a scheme which is of order 2.
Paper Structure (9 sections, 3 theorems, 100 equations, 1 figure, 2 algorithms)

This paper contains 9 sections, 3 theorems, 100 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Stochastic Heat Equation
  3. Stochastic wave equation
  4. Numerical simulations
  5. Mathematical setup and preliminarily results
  6. Stochastic Heat Equation
  7. Modified Crank--Nicolson Scheme
  8. Stochastic Wave Equation
  9. Modified Crank--Nicolson scheme for SPDE \ref{['eq:wave']}

Key Result

Lemma 2.1

Let $(\mathrm{B}, \|\cdot\|_{\mathrm{B}})$ be a Banach space. Assume $f \in C^{1,\gamma}([0,T]; \mathrm{B})$ for some $\gamma \in (0,1]$and there exists a constant $\widetilde{C}>0$ such that Then for any interval $[a, a+\kappa] \subset [0,T]$ we have that

Figures (1)

  • Figure 1: Convergence rates for the stochastic heat equation \ref{['eq:heat']} (left) and wave equation \ref{['eq:wave']} (right), averaged over $1000$ Monte Carlo realizations.

Theorems & Definitions (6)

  • Remark 1.1
  • Lemma 2.1
  • Theorem 3.1: first main result
  • proof
  • Theorem 4.1: second main result
  • proof